Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.     

Analytical modeling of synthetic fiber ropes subjected to axial loads. Part I: A new continuum model for multilayered fibrous structures

Synthetic fiber ropes are characterized by a very complex architecture and a hierarchical structure. Considering the fiber rope architecture, to pass from fiber to rope structure behavior, two scale transition models are necessary, used in sequence: one is devoted to an assembly of a large number of twisted components (multilayered), whereas the second is suitable for a structure with a central straight core and six helical wires (1 + 6). The part I of this paper first describes the development of a model for the static behavior of a fibrous structure with a large number of twisted components. Tests were then performed on two different structures subjected to axial loads. Using the model presented here the axial stiffness of the structures has been predicted and good agreement with measured values is obtained. A companion paper (Ghoreishi, S.R. et al., in press. Analytical modeling of synthetic fiber ropes, part II: A linear elastic model for 1 + 6 fibrous structures, International Journal of Solids and Structures, doi:10.1016/j.ijsolstr.2006.08.032) presents the second model to predict the mechanical behavior of a 1 + 6 fibrous structure.     

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

An efficient scheme, called quasi-linearization finite differences, is developed for large-deflection analysis of prismatic and non-prismatic slender cantilever beams subjected to various types of continuous and discontinuous external variable distributed and concentrated loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender cantilever beams with different loading conditions are analyzed to illustrate the merits of the adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large-displacement finite element analysis by the MSC/NASTRAN program. A comparison between the present secheme, MSC/NASTRAN and available results from the literature reveals excellent agreement. The advantage of the new scheme is that the load can be applied in one step with few iterations (3–6 iterations).     

Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet

This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet.